Finite Group Databasebeta

← #366923

Subgroups — #366923

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1939C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7C4⋊C42C211yesmaximal, characteristic, complemented
8#11358C212yesmaximal, complemented
9#11076C212yesmaximal
10#10931C212yesmaximal
11#10850C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2563—24no—
21#2563C418yescomplemented
22C2×#393—22no—
23C2×#390C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26C2×#380—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1990—24no—
33#1987—24no—
34#1939—24no—
35#1802—24no—
36C4×#102—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2212yes—
45#1436—22no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—42no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97C2×#102Q814yescomplemented
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101#393—216no—
102#390—24no—
103#390C2×C418yes—
104#384—22no—
105#384D414yes—
106#382—24no—
107#382C2×C418yes—
108#380—28no—
109#380—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—42no—
113C4×C4⋊C4—44no—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4D414yes—
116C4×C4⋊C4D414yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—22no—
130#340—22no—
131#336—24no—
132C25—21no—
133C22×Q8—24no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—81no—
137C22×Q8—81no—
138C22×Q8—81no—
139C22×Q8—81no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C22×D4C2×Q812yes—
153C22×D4#4112yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×Q812yes—
191C22×C8#4112yes—
192C4⋊Q8—28no—
193C4⋊Q8—28no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C4×D4—28no—
197C4×D4—28no—
198C4×D4—44no—
199C4×D4—44no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4C22⋊C414yes—
237C2×C4⋊C4C22⋊C414yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×Q812yes—
244C2×C4⋊C4#4112yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4—82no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—82no—
264C2×C42C22×C414yes—
265C2×C42C22×C412yes—
266C2×C42C2×D414yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42#4112yes—
271C2×C42#4112yes—
272C4⋊C8—48no—
273C4⋊C8—44no—
274#102—28no—
275#102—28no—
276#102—48no—
277C4×C8—44no—
278#95—28no—
279#95—28no—
280#95—24no—
281#95—44no—
282#95—44no—
283#95—82no—
284#95C22⋊C418yes—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24C22×D411yescharacteristic
301C2×Q8—48no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4C4×Q814yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#13512yes—
403C22×C4#13512yes—
404C22×C4AGL(1,ℤ/8)12yes—
405C22×C4#13712yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×Q811yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C42−1+411yescharacteristic
418C22×C42−1+411yescharacteristic
419C22×C42−1+411yescharacteristic
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—84no—
465C4⋊C4C22⋊Q814yes—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42—24no—
473C42—48no—
474C42—88no—
475C42C4×D414yes—
476C42C4×D414yes—
477C42#12414yes—
478C42#12714yes—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23C22×C22⋊C411yescharacteristic
515C23C2×#12111yescharacteristic
516C23C2×#12111yescharacteristic
517C23#54611yescharacteristic
518C23#54711yescharacteristic
519C23#54811yescharacteristic
520C23D4×Q811yescharacteristic
521C23C2×#13511yescharacteristic
522C23C2×#13511yescharacteristic
523C23C2×AGL(1,ℤ/8)11yescharacteristic
524C23C2×#13711yescharacteristic
525C23#57611yescharacteristic
526C23#57611yescharacteristic
527C23#57711yescharacteristic
528C23#57711yescharacteristic
529Q8—816no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—816no—
594C2×C4#39014yes—
595C2×C4#39014yes—
596C2×C4#41614yes—
597C2×C4#41714yes—
598C2×C4#46314yes—
599C2×C4#46314yes—
600C2×C4#46414yes—
601C2×C4#46414yes—
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×D411yescharacteristic
604C2×C4C2×C4×Q811yescharacteristic
605C2×C4C4×#4111yescharacteristic
606C2×C4#51911yescharacteristic
607C2×C4#52011yescharacteristic
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4C2×#12411yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4C2×#12711yescharacteristic
613C2×C4#53311yescharacteristic
614C2×C4#53611yescharacteristic
615C2×C4#53711yescharacteristic
616C2×C4#54111yescharacteristic
617C2×C4#54211yescharacteristic
618C2×C4#54611yescharacteristic
619C2×C4#54611yescharacteristic
620C2×C4#54711yescharacteristic
621C2×C4#54711yescharacteristic
622C2×C4#54811yescharacteristic
623C2×C4#54811yescharacteristic
624C2×C4D4×Q811yescharacteristic
625C2×C4D4×Q811yescharacteristic
626C2×C4#55211yescharacteristic
627C2×C4#55211yescharacteristic
628C2×C4#55411yescharacteristic
629C2×C4#55611yescharacteristic
630C8—42no—
631C8—42no—
632C8—82no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22C2×#39011yescharacteristic
649C22#233911yescharacteristic
650C22Q8×C22⋊C411yescharacteristic
651C22#241511yescharacteristic
652C22#241711yescharacteristic
653C22#242711yescharacteristic
654C22#242911yescharacteristic
655C22C2×#41611yescharacteristic
656C22C2×#41711yescharacteristic
657C22#289711yescharacteristic
658C22#289711yescharacteristic
659C22C2×#46311yescharacteristic
660C22C2×#46311yescharacteristic
661C22C2×#46411yescharacteristic
662C22C2×#46411yescharacteristic
663C22#304111yescharacteristic
664C22#304311yescharacteristic
665C22#304511yescharacteristic
666C22#304511yescharacteristic
667C22#330311yescharacteristic
668C22#330511yescharacteristic
669C22#332511yescharacteristic
670C22#332911yescharacteristic
671C22#334011yescharacteristic
672C22#334111yescharacteristic
673C22#335411yescharacteristic
674C22#335711yescharacteristic
675C22#336011yescharacteristic
676C22#336111yescharacteristic
677C22#336611yescharacteristic
678C22#337211yescharacteristic
679C22#338011yescharacteristic
680C22#338111yescharacteristic
681C22#338411yescharacteristic
682C22#338511yescharacteristic
683C4—24no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—84no—
689C4—82no—
690C4—84no—
691C4#198318yes—
692C4C4×C22⋊Q811yescharacteristic
693C4#232311yescharacteristic
694C4#236011yescharacteristic
695C4#236211yescharacteristic
696C4#248911yescharacteristic
697C4#249211yescharacteristic
698C4#254011yescharacteristic
699C4#254211yescharacteristic
700C2—84no—
701C2#1399711yescharacteristic
702C2C2×#198311yescharacteristic
703C2#2126311yescharacteristic
704C2#2248911yescharacteristic
705C2#2250311yescharacteristic
706C2#2251911yescharacteristic
707C2#2252211yescharacteristic
708C2#2467711yescharacteristic
709C2#2467911yescharacteristic
710C2#2481211yescharacteristic
711C2#2481411yescharacteristic
712C2#2531911yescharacteristic
713C2#2532111yescharacteristic
714C2#2545011yescharacteristic
715C2#2545211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1987C2semidirect
4. C2×#1939C2semidirect
7. C4⋊C42C2semidirect
8. #11358C2semidirect
11. #10850C2semidirect
21. #2563C4semidirect
97. C2×#102Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25632
C2×#19902
C2×#19872
C2×#19392
C2×#18022
C2×C4×#1022
C4⋊C422
#113582
#110762
#109312
#108502